Money and Price Posting Under Private Information!

Money and Price Posting under Private Information

Mei Dongy Janet Hua Jiangz

September 16, 2010

Abstract

We study price posting with undirected search in a search theoretic model with divisible

money and divisible goods. Ex ante homogeneous buyers experience match speci…c preference

shocks in bilateral trades. The shocks follow a continuous distribution and the realization

of the shocks is private information. We show that generically there exists a unique price

posting monetary equilibrium. In equilibrium, each seller posts a continuous pricing schedule

that exhibits quantity discounts. Buyers spend only when they have high preferences, and as

preferences are higher, they spend more. As in‡ation decreases the value of waiting for future

trades, higher in‡ation induces more buyers to spend money and those who spend before to

spend their money faster. The model naturally captures the hot potato e¤ect of in‡ation

along both the intensive margin and the extensive margin.

Keywords: Price posting, Undirected search, Private information, Hot potato e¤ect, In‡a-

tion, Quantity discounts

JEL: D82, D83, E31

We thank Aleksander Berentsen, Jose Dorich, Huberto Ennis, Ben Fung, Peter Norman, Randy Wright, and seminar participants at the Bank of Canada, the 2009 Canadian Economic Association Meetings in Toronto, the 2009 Chicago Fed Summer Workshop on Money, Banking and Payments, the 2010 Midwest Macro Meetings, the 2010 Econometric Society World Congress in Shanghai, and the University of Manitoba Delta Marsh Conference on Monetary Economics for their comments and suggestions. The views expressed in this paper are those of the authors. No responsibility for them should be attributed to the Bank of Canada. ythe Bank of Canada; email: [email protected] zDepartment of Economics, University of Manitoba; email: [email protected]

1 1 Introduction

This paper builds a search theoretic monetary model based on the framework developed by Lagos and Wright (2005) and Rocheteau and Wright (2005), where sellers post prices and buyers engage in undirected search in the sense that buyers only see the posted prices after they are matched with sellers. Price posting with undirected search (hereafter price posting) is an interesting pricing mechanism to study because it captures the characteristics of many daily exchanges.1 In many occasions, buyers randomly enter a store, read the prices and decide whether or not to make a purchase.

In the search theoretic monetary literature, very few papers have studied price posting because it is hard to generate equilibria with valued …at money under this pricing mechanism. Generally, the existence of monetary equilibria hinges critically on the condition that the buyer extracts some surplus through trading with the seller. In a typical monetary model with price posting, sellers can extract all the trading surplus by proposing prices and quantities, and buyers decide whether to trade or not. In this case, sellers have all the bargaining power so that in equilibrium no buyers want to hold money when the nominal interest rate is positive. Monetary equilibrium thus unravels under price posting.

To generate monetary equilibria under price posting, we introduce match speci…c preference shocks which a¤ect buyers’ marginal willingness to consume and private information about the realization of the shocks. When the preference shocks follow a continuous distribution, we show that there exists a unique monetary equilibrium.

The model is then used to examine the seller’s pricing schedule, the buyer’s spending pattern and the e¤ects of in‡ation. We …nd that in equilibrium, sellers post a non-linear pricing schedule and the unit price exhibits quantity discounts: the unit price is lower for purchases of larger sizes.

Quantity discounts are frequently observed in practice and the traditional explanation is that the unit cost of producing a large quantity is lower. In our model, quantity discounts exist even when the unit cost is constant. It is used by sellers to induce buyers with higher preferences to consume more. Buyers spend money only when they have high enough marginal utilities and spend more when their marginal utilities are higher.

As for the e¤ects of in‡ation, we show that buyers spend their money faster when in‡ation is higher; this is the "hot potato" e¤ect of in‡ation. The intuition is that in‡ation reduces the bene…t of retaining money for future trading opportunities and induces buyers to spend their money faster.

1 We use price posting to refer to price posting with undirected search in the rest of the paper. When price posting is combined with directed search, it is generally labelled as competitive search in the literature.

2 More importantly, price posting equilibrium endogenously generates the hot potato e¤ect along both the intensive (those who spend choose to spend more) and the extensive margins (those who do not spend start spending) without resorting to free entry or search intensity.

There have been a few papers that study price posting. As in our paper, the key to the existence of a monetary equilibrium under price posting is private information about match speci…c preference shocks. Jafarey and Masters (2003) and Curtis and Wright (2004) are built on the indivisible money model of Trejos and Wright (1995). In Curtis and Wright (2004), there are K 2 realizations of the preference shock and in equilibrium, sellers post at most two prices. In Jafarey and Masters (2003), the preference shock follows a uniform distribution. Interestingly, there is only a single price posted in equilibrium. In general, there exists the law of two prices in models with indivisible money as emphasized by Curtis and Wright (2004). More recently, Ennis (2008) extends price posting to a divisible money framework as in Lagos and Wright (2005). He speci…es a distribution of preference shocks that takes two values, and shows that sellers post a single price in equilibrium.

The above three papers generate monetary equilibria under price posting, however, they are not suitable to analyze sellers’pricing behavior and buyers’spending pattern and how they are a¤ected by in‡ation. In equilibrium, at most two prices are posted, and buyers spend all their money during exchanges. In our model, money is divisible and preference shocks follow a continuous distribution.

The model implies richer (and more realistic) pricing behavior and allows us to study the e¤ect of in‡ation on buyers’spending pattern in a natural environment.

Several recent papers have tried to rationalize the hot potato e¤ect of in‡ation using microfounded monetary theory. Lagos and Rocheteau (2005) endogenize search intensity, and in‡ation induces buyers to search more intensively. The explanation, however, is not robust and only applies to competitive search at low in‡ation. In Ennis (2009), sellers have more opportunities to rebalance money holdings than buyers, in‡ation makes buyers search harder to …nd sellers to o¤-load their cash. Liu et al. (forthcoming) o¤er another explanation. They assume that buyers pay the entry fee to enter the market. In‡ation reduces buyers’trading surplus, so fewer buyers choose to enter, which increases the matching probability for those who do enter. Nosal (forthcoming) assumes that accepting a current trade reduces the probability of future trading. In‡ation reduces the value of future trading and buyers are more likely to accept current trades.

The rest of the paper proceeds as follows. Section 2 describes the environment. In Section 3, we characterize the monetary equilibrium when the preference shock follows a uniform distribution.

Section 4 examines the e¤ect of in‡ation and rationalizes the hot potato e¤ect of in‡ation. In

Section 5, the model is extended to allow for a more general continuous distribution of the preference

3 shock. Finally, Section 6 concludes. The technical proofs of results in Section 5 are provided in the

Appendix.

2 Environment

The model is based on Lagos and Wright (2005). Time is discrete and runs from 0 to .A 1 decentralized market (DM) and a centralized market (CM) open sequentially in each period. The discount factor between two periods is 0 < < 1. There are two permanent types of agents: buyers and sellers distinguished by their roles in the DM. There is one nonstorable good in each submarket: a CM good and a DM good.

The CM is a centrally located competitive spot market. In the CM, all agents can consume or produce the CM good x. The utility of consuming x units of the CM good is x. If x < 0, it means that an agent produces x units of the CM good and incurs disutility x. In the DM, agents are anonymous. Buyers and sellers are randomly matched so that one buyer meets one seller with probability 1. Buyers are those who want to consume but cannot produce. Sellers, on the other hand, can produce but do not want to consume. This generates a lack of double coincidence of wants problem. Together with anonymity, money becomes essential as the medium of exchange.

For a seller, the disutility of producing q units of the DM good is c(q) with c(0) = 0, c0 > 0, and c00 0. By consuming q units of the DM good, a buyer’sutility is eu(q) where e 0 is a preference parameter that determines the buyer’s marginal utility of consumption. The utility function u(q) satis…es u(0) = 0, u0 > 0 > u00 and u0(0) = . 1 All buyers are ex ante identical before being matched with sellers. Ex post, however, they are subject to match speci…c preference shocks and become heterogeneous during their matches with sellers. The realization of e follows a uniform distribution on the interval [0; 1]. Buyers hold private information about the realization of e.

The terms of trade in the DM are characterized by price posting with undirected search. Before a buyer and a seller are matched, the seller posts terms of trade which consist of a menu of price- quantity pairs, and the buyer does not observe the posting. Once they are matched, the buyer sees the posted terms of trade and decides which price-quantity pair to take from the menu. As in other papers that study price posting, buyers may choose not to trade at all after they are matched.2

Figure 1 describes the timeline of events.

2 Since agents are anonymous in the DM, they are perceived to lack commitment. Assuming that buyers can choose not to trade is consistent with the environment where there is a lack of commitment.

4 Fiat money is supplied by the monetary authority. Money supply grows at a constant rate so that Mt = Mt 1. New money is used to …nance a lump-sum transfer to buyers at the beginning of each CM. Let Tt = ( 1)Mt 1 be the amount of nominal transfer to each buyer. CM DM buyer

e realized; choose the terms of trade work; choose $ and consume

t t+1 consume post terms produce of trade

seller

Figure 1: Timeline of Events

3 Price-Posting Equilibrium

Throughout this paper, we assume that money balances are observable. To solve the equilibrium, we begin with analyzing agents’choices in the CM and then move to consider the DM.

3.1 Decision Making in the CM

In the CM, agents rebalance their money holdings by trading money for the CM good x or vice versa. We …rst consider the buyer’s problem. Let W b(m) denote a buyer’s value function while entering the CM with m units of money. Let m^ + be the money holding upon entering into the DM

b in the next period, and V (m ^ +) be the associated value function. The buyer’svalue function is

b b W (m) = max x + V (m ^ +) x;m^ + s.t. x = (m + T m^ +):

5 where is the value of money in the CM. De…ning z = m, = T and z^+ = m^ +; we can rewrite the buyer’sproblem in real terms as

b b W (z) = max z + z^+ + V (^z+) . z^+ +

b Note that due to quasilinearity, the choice of z^+ is independent of z and W (z) is linear in z, where dW b(z)=dz = 1. The …rst order condition is

b dV (^z+) = if z^+ > 0: dz^+ +

For a seller with z units of real money balance upon entering the CM, let W s(z) be his value function. The seller simply spends all money that they accumulated in the previous DM and carry 0 money balance to the next DM. The seller’svalue function is W s(z) = z + V s(0) and dW s(z)=dz =

3.2 Decision Making in the DM

In the DM, the seller can potentially post a schedule of price-quantity pairs (qe; ze) for all e [0; 1]. 2 We assume that the seller can observe z^, the buyer’schoice of money balance to enter into the DM, so that the seller’sposting may depend on z^.3 The matching function is such that each buyer meets a seller. Upon matching, the buyer realizes the match speci…c preference shock and the shock is the buyer’sprivate information. After seeing the seller’sposting, the buyer decides whether to trade or not, and if trade, which (qe; ze) to take. The seller’schoice of pricing schedule is in essence a mechanism design problem, where the seller is the principal and the buyer is the agent with private information (about their preferences). We can apply the revelation principle to formulate the seller’s problem as follows: taking the buyer’s z^, and the distribution of the preference shocks as given, choose (qe; ze)e [0;1] to maximize his DM 2 3 Refer to Ennis (2008) for a detailed discussion of this assumption.

6 value function. Formally,

1 s s V (0) = max [ c(qe) + W (ze)]de qe;ze e [0;1] 0 f g 2 Z

qe 0, (NN) 8 > > z z^, (CC) > e s.t. > > > < eu(qe) ze eu(qe ) ze for all e; e0 [0; 1], (IC) 0 0 2 > > eu(qe) ze 0. (PC) > > > :> The four constraints are the non-negativity constraints, the cash constraints, the incentive constraints and the participation constraints, respectively. Notice that we use the property that W b(z) is linear in z to derive the IC. To solve the seller’soptimization problem with private information, we use the result from Mas-Collell, Winston and Green (1995, Proposition 23.D.1, page 888) to …nd the necessary and su¢ cient conditions to guarantee that the incentive constraints are satis…ed. Then we replace the incentive constraints with these conditions and convert the seller’s problem into an optimal control problem.4

Let ve eu(qe) ze denote the buyer’s ex post trading surplus when the realization of the preference shock is e. The solution (qe; ze) is incentive compatible if and only if

@qe=@e 0, e e ve = v xdx = u(qx)dx or v e = u(qe). Z0 Z0

We can ignore the PCs for all e > 0 because they are implied by the PC for e = 0 and the ICs. In addition, the PC binds for buyers with e = 0. The seller’s problem can be restated as an optimal

4 Faig and Jerez (2006) use a similar technique in a model where terms of trade are determined by competitive search and buyers have private information about the realization of preference shocks.

7 control problem with ve as the state variable and qe as the control variable,

1 max [ c(qe) + eu(qe) ve]de qe;ve e [0;1] 0 f g 2 Z

qe 0, (NN) 8 > > eu(qe) ve z^, (CC) > > s.t. > > @qe=@e 0, (IC1) > <> v e = u(qe), (IC2) > > > > v = 0. (PC) > 0 > > :> To proceed, we …rst disregard the constraint @qe=@e 0 (and we will impose it later). The Hamiltonian of the optimal control problem is

H = [ c(qe) + eu(qe) ve] + eu(qe);

where e is the co-state variable. The Lagrangian is

= c(qe) + eu(qe) ve + eu(qe) L

+ [^z eu(qe) + ve] + eqe; e

where e and e are the Lagrangian multipliers associated with the cash constraint and the non- negativity constraint, respectively. The …rst-order conditions are

qe :(e + e e)u0(qe) = c0(qe) e; (1) e ve : 1 + = e; (2) e

e : u(qe) = v e; (3)

:z ^ eu(qe) + ve 0; if > 0, then = 0; (4) e e

e : qe 0; if > 0, then e = 0. (5)

The transversality condition is 1v1 = 0. For monetary equilibrium to exist, v1 > 0 and 1 = 0.

8 Integrating e over the interval [e; 1], we have

1 1 1 1 xdx = (1 )dx 1 e = 1 e dx e = (e 1) + e where e dx: x ! x ! x Ze Ze Ze Ze where the last step uses the transversality condition. It follows that (1) becomes

[(2e 1) + e e]u0(qe) = c0(qe) e (6) e

Now we impose the constraint @qe=@e 0 (and as a result @ze=@e 0). Given this, we can consider the seller’sproblem in three regions of e divided by 0 e0 e^ 1. Case (a). When e is small (e e0), no exchange occurs or qe = ze = ve = 0. The NN binds and the CC is loose. In this case, q is constant at 0 so the constraint @qe=@e 0 is satis…ed. Case (b). For intermediate values of e (e0 e e^), neither NN nor CC binds, i.e., = e = 0. e Hence, (6) reduces to5

[(2e 1) + e^]u0(qe) = c0(qe), (7) which can be used to solve qe. Since u00 < 0 c00, we can conclude that @qe=@e > 0. As a result, the constraint @qe=@e 0 is automatically satis…ed. Case (c). When e is high (e e^), the buyer is charged all his money holdings, or ze =z ^ and qe =q: ^ The constraint @qe=@e 0 is again satis…ed and we can solve q^ from

[(2^e 1) + e^]u0(^q) = c0(^q). (8)

In the next step, we will …nd the term e^ in (7) and (8) as a function of e^. Since qe =q ^ for all e > e^, q^ also solves

[2e 1 + e e] u0(^q) = c0(^q): (9) e : Combining (8) and (9), and using = e, we have a di¤erential equation of e; e

: 2^e + e^ = 2e + e + ee. (10)

5 Notice that since = 0 for e [0; e^], we have e = e^ for e [0; e^]. e 2 2

9 6 Together with the end-point conditions e^ = 0 and 1 = 0, (10) gives rises to the following result

2 1 e = (1 e) +e ^ 1 , e 2 e^ = (1 e^) , (11) e^ = 1 ( )2 . e e

The …nal step to complete the solution to the seller’s problem is to determine e0 and e^. First, note that e0 can be expressed as a function of e^ and is determined by

e^2 [(2e0 1) + e^]u0(0) = c0(0) or e0 =e ^ . 2

7 It then su¢ ces to …nd e^ as a function of z^. From IC2 and the de…nition of ve, e^ is solved from

e^2 1 z^ = (^e )u(^q) + c(^q): 2 2

In general, it is possible that e^ takes the corner solution e^ = 1 (so that no buyer is cash con- strained) when z^ is large enough. This situation occurs when z^ z with z given by

1 z = [u(q ) + c(q )] and u (q ) = c (q ). 2 1 1 0 1 0 1

In this situation, buyers with highest preferences (e = 1) buy q1 amount of goods with z units of money.

To summarize, the seller’s pricing schedule (qe; ze) and ve are functions of z^. Lemma 1 states that the pricing schedule posted by the seller is unique and is a function of z^.

Lemma 1 For any given z^ z, the optimal solution (qe; ze; ve) for all e [0; 1] to the seller’s price 2 posting problem is unique and is characterized by:

(i) For e [0; e0], qe = ze = ve = 0; 2 6 k 2 We solve (10) by a guess-and-verify method. We guess that e = 1 + es and …nd that k = e^ . 7 e^ e^ Integrating both sides of IC2 with respect to qe over [0; e^] gives 0 [2e 1 + e^]u0(qe)dqe = 0 c0(qe)dqe e^ or 0 2eu0(qe)dqe (1 e^)u(^q) = c(^q). Implementing integratingR by parts on the …rst termsR gives e^ e^ 2 eu^ (^q) u(qe)de (1 e^)u(^q) = c(^q). Combine this with ve = u(qe)de =eu ^ (^q) z^ and we have the R 0 0 equationh below.R i R

10 (ii) For e [e0; e^], 2

2 qe : (2e 2^e +e ^ )u0(qe) = c0(qe); (12) e^2 1 ve = e e^ + u(qe) c(qe); 2 2 e^2 1 ze = e^ u(qe) + c(qe); (13) 2 2

(iii) For e [^e; 1], 2

2 qe =q ^ :e ^ u0(^q) = c0(^q); (14)

ze =z; ^ (15)

ve = eu(^q) z^;

(iv) e0 and e^ are given by

e^2 1 z^ = (^e )u(^q) + c(^q) (16) 2 2 e^2 e0 =e ^ (17) 2

For z^ > z, (16) is replaced by e^ = 1:

3.3 Monetary Equilibrium

Given the seller’s pricing schedule in the DM, we are now ready to derive the buyer’s demand for money in the CM. As the buyer knows how (qe; ze; ve) depends on z^, the buyer’s value function in the DM with z^ is given by

1 b b V (^z) = [eu(qe(^z)) +z ^ ze(^z)]de + W (0) Z0 = S(^z) +z ^ + W b(0);

1 where S(^z) ve(^z)de is the expected trading surplus in the DM. In the CM, the buyer choice of 0 z^+ satis…es R b @V (^z+) = = [S0 (^z+) + 1] : + @z^+

11 We will focus on the steady-state equilibrium where z^ is constant and =+ = . The equilibrium z^ is given by

1 = S0(^z). (18)

The nominal interest rate follows the Fisher equation and is given by i = = 1. As long as > or the nominal interest rate is positive, the cash constraint will bind for buyers with high enough preferences, so the optimal z^ cannot be greater than z. The buyer’schoice of z^ from (18), together with the seller’s optimal pricing schedule, completes the description of the price posting monetary equilibrium.

De…nition 1 In the steady state, a price-posting monetary equilibrium is a list of (qe; ze) for all

s e [0; 1] and z^ such that [1] (qe; ze) maximize V (0) given z^ (characterized by (12),(13),(14),(15), 2 (16) and (17)) ; and [2] z^ maximizes W b(z) for any z (characterized by (18)).

Proposition 1 There exists a unique monetary equilibrium for generic values of .

Proof. We …rst show that knowing how (qe; ze) depends on z^, buyers choose a unique z^ in the CM. Note that

1 2 dq^ 1 2 de^ S0(^z) = (1 e^ )u0(^q) (1 e^) e^ u(^q) c(^q) (1 e^); 2 dz^ 2 dz^ where dq=d^ z^ and de=d^ z^ are solved by di¤erentiating (14) and (16) with respect to z^. We can see that

S0(^z) does not depend on i directly. Following the proof in Wright (2008), z^ is unique for generic values of . It follows from proposition 1 that a unique z^ leads to a unique pricing schedule. In addition, one can show that S0(0) = . Hence, monetary equilibrium exists and it is unique for 1 generic values of . In price posting equilibrium, all buyers choose the same z^ in the CM. As long as < , we 1 have z^ > 0 and e;^ e0; (^e e0) > 0. A continuum of prices is observed when preference shocks follow a uniform distribution. The number of prices observed in equilibrium is measured by e^ e0. This is in contrast to Ennis (2008), where only a single price is observed with a two-point distribution of preference shocks. This is also in contrast to Jafarey and Masters (2003), and Curtis and Wright

(2004), where there can be at most two prices in equilibrium because money is indivisible. Therefore, the law of two prices cannot be generalized to models with divisible money. Compared with the previous papers on price posting, our model generates a richer pricing schedule (on the seller’sside) and spending pattern (on the buyer’sside).

When buyers hold private information about match speci…c preference shocks that follow a uniform distribution, sellers post prices such that the pricing schedule can di¤erentiate buyers of

12 di¤erent ex post preferences. Buyers with e [0; e0] choose not to spend money and consume 2 nothing. Buyers in this group do not value consumption much. As a result, it is better for them to hold on to their money balances and wait for future consumption opportunities. For buyers with e [^e; 1], the cash constraint binds and they exhaust their money holdings during exchanges. Buyers 2 with e [e0; e^] spend part of their money holdings and as e increases, qe and ze both increase. In 2 equilibrium, a positive measure (^e e0) of prices is observed. The unit price is given by

2 ze e^ u(qe) 1 c(qe) = e^ + for e [e0; e^]: (19) q 2 q 2 q 2 e e e

In general, ze=qe depends on qe, which means that the pricing schedule is nonlinear. The next proposition establishes that larger quantities are associated with lower per unit prices when the cost function is linear.

Proposition 2 Quantity discounts: when c(q) = q, d(ze=qe)=dqe < 0 for e [e0; e^]. 2

The proof follows directly from (19). This result appears to be consistent with the pricing strategy in reality. One common practice used by sellers is to o¤er quantity discounts, i.e., lower unit prices for larger purchases. The traditional explanation for quantity discounts is that the unit cost for sellers to produce larger quantities is smaller. Here, we …nd that even if the unit cost is constant, sellers may still choose to o¤er quantity discounts. The model provides an alternative explanation for quantity discounts. When buyers’preferences are private information, sellers o¤er quantity discounts to induce buyers with higher preferences to buy more.

One may wonder if quantity discounts still hold when the cost function is convex. In this case, there are two factors that a¤ect the per unit price – convex cost and private information. The convex cost function implies that per unit price tends to increase when the quantity becomes larger.

In Peterson and Shi (2004), buyers experience match speci…c preference shocks. In each bilateral trade, price is determined by the buyer’s take-or-leave-it o¤er. The model generates a continuous distribution of prices. Under the assumption of a convex cost function (a standard assumption adopted in the monetary search literature), the model predicts that higher quantities are associated with higher per unit prices.8 With a convex cost function, it is not obvious analytically whether the per unit price becomes higher or lower as quantity increases. From the numerical exercise, quantity discounts hold even when the cost function is very convex. It seems that the e¤ect of private information on per unit price dominates the e¤ect of the convex cost function.

8 It can be shown that competitive pricing with bilateral trades generates a similar pricing pattern.

13 4 Hot Potato E¤ect of In‡ation

In this economy, it is easy to verify that the Friedman rule cannot achieve the …rst-best allocation.

In‡ation a¤ects the economy along both the extensive and intensive margins. The extensive margin is captured by e0, the measure of buyers who choose not to trade ex post. The intensive margin is captured by qe for e [e0; 1]. 2

Proposition 3 The e¤ects of in‡ation on (^z; e0; e;^ e^ e0): dz=d ^ < 0; de0=d < 0; de=d ^ < 0 and d(^e e0)=d < 0.

Proof. Consider z^ =z . As e^ = 1, it is straightforward that S0(z) = 0. The optimal z^ satis…es

S0(^z) + 1 = 0. Since we focus on z^ z, it implies that S0(z) + 1 < 0 for > and ! from above. As S0(0) = , we know that S0(0) + 1 = . For z^ [0; z], a generically unique 1 1 2 dz^ 1 optimal z^ implies that S00(^z) < 0. From S0(^z) + 1 = 0, = < 0. d S00(^z) dq^ de^ de^ Using (14), one can …nd d in terms of d , which is substituted into (16) to get d < 0. It

de0 d(^e e0) follows from (17) that d < 0 and d < 0 as well.

Proposition 4 The e¤ects of in‡ation on (qe; ze; ze=z^) for e [e0; e^]: dqe=d > 0, dze=d > 0, and 2 d(ze=z^)=d > 0 if u000 0 and c000 0.

Proof. As de=d ^ < 0, proving dqe=d > 0 and dze=d > 0 is equivalent to proving dqe=de^ < 0 and

dqe dze dze=de^ < 0. We do this by proving d de =de^ < 0 and d de =de^ < 0, or qe(e;e ^) and ze(e;e ^) are steeper for lower values of e^.

dqe We …rst show d =de^ < 0. Use (12) that characterizes qe for e [e0; e^] to calculate dqe=de de 2 as (to simplify notations, we omit the arguments of the functions u( ) and c( )):

dq 2u 2 e = 0 de c u c u 00 0 0 00

2 (u0) The term c u c u > 0 decreases in e^ if c000 0 and u000 0: 00 0 0 00

2 2 d u0 d u0 c00u0 c0u00 c00u0 c0u00 dqe = de^ dqe de^ 2 2u0(c00u0 c0u00) u0 (c000u0 + c00u00 c00u00 c0u000) dqe _ 2 (c00u0 c0u00) de^ 3 2 2u0(c00u0 c0u00) u0 c000 + u0 c0u000 dqe = (c u c u )2 de^ 00 0 0 00 < 0 if c000 < 0 and u000 > 0:

14 dze Now we show d de =de^ < 0. Recall that dze=de = eu0(qe)dqe=de, so dz eu 3 e = 0 > 0: de c u c u 00 0 0 00

3 eu0 The term c u c u decreases in e^ if c000 0 and u000 0 because 00 0 0 00

3 eu0 d c u c u 00 0 0 00 dqe 2 3 3u0 (c00u0 c0u00) u0 (c000u0 + c00u00 c00u00 c0u000) = e 2 (c00u0 c0u00) 2 4 3 3u0 (c00u0 c0u00) u0 c000 + u0 c0u000 = e (c u c u )2 00 0 0 00 > 0 if c000 < 0 and u000 > 0:

As dze=d > 0 and dz=d ^ < 0, d(ze=z^)=d > 0. Here, we have the standard result that in‡ation decreases the demand for real money balances.

Besides that, the model provides a natural explanation about the "hot potato" e¤ect of in‡ation, which says that buyers choose to spend money faster when in‡ation is higher. In price posting equilibrium, buyers spend money only when their preferences are high enough, and those with higher preferences spend more. In‡ation reduces the future purchasing power of money and the bene…t of retaining money, so buyers spend money faster. The model can capture the hot potato e¤ect along both the extensive and intensive margins. As shown in Proposition 3, in‡ation reduces e0, implying that more buyers start spending money. Proposition 4 implies that for those who spend money, higher in‡ation induces them to spend more money, captured by higher ze=z^. Refer to Figure 2 for a graphical illustration of the hot potato e¤ect.

Several papers in monetary theory have tried to rationalize the hot potato e¤ect. Lagos and

Rocheteau (2005) endogenize search intensity, and they show that in‡ation may increase buyer’s trading surplus and their search intensity. However, the result holds only for low in‡ation rates when the pricing mechanism is competitive search. Bargaining cannot deliver similar results because in‡ation monotonically reduces the buyer’s surplus in a match and thus the search e¤ort. Ennis

(2009) assumes that sellers have more opportunities to rebalance money holdings than buyers (i.e., they have more frequent access to the centralized market than buyers), in‡ation makes buyers search harder to …nd sellers to o¤-load their money.

15 Hot potato effect

High inflation Low inflation

0 e h l h l 1 e0 e0 eÖ eÖ

z e

High inflation

Low inflation

0 e h l h l 1 e0 e0 eÖ eÖ

ze / zÖ

High inflation

Low inflation

0 e h l h l 1 e0 e0 eÖ eÖ

Figure 2: Hot Potato E¤ect of In‡ation

More recently, Liu et al. (forthcoming) assume that buyers pay the entry fee to enter the market.

In‡ation reduces buyers’trading surplus, so fewer buyers choose to enter. For those who do enter, the matching probability is higher. Buyers are able to spend faster simply because they have more opportunities to trade, not because they actively try to get rid of their money. Furthermore, the result that the matching probability is higher for buyers when the in‡ation rate is higher depends on whether free entry is assumed by buyers or sellers. For example, Rocheteau and Wright (2005) allow free entry by sellers and in‡ation monotonically reduces buyers’trading probability. To explain the hot potato e¤ect of in‡ation, it is not clear that one should assume free entry by buyers rather than by sellers.

In Nosal (forthcoming), it is assumed that accepting a current trade involves an exogenous cost that lowers the probability of future trading. In‡ation reduces the value of future trading and people are more likely to accept a current trade. Our paper and Nosal (forthcoming) both emphasize that buyers spend faster in fear that money will lose purchasing power in the future due to in‡ation. We

16 provide a more integrated model and do not rely on assuming the exogenous reduction of future trading probability as a consequence of accepting a current trade.9

5 Extension to More General Distributions

The above analysis assumes that the distribution of preference shocks follows a uniform distribution.

In this section, we extend our model by allowing preference shocks to follow a general continuous distribution. Let f(e) and F (e) denote the p.d.f. and the c.d.f. of the distribution, respectively.

Without loss of generality, we still focus on e [0; 1]. We assume that ef(e) is increasing in e, i.e., 2 f(e) + ef 0(e) > 0.

5.1 Seller’sPrice Posting

The choice problems for buyers and sellers in each market remain the same as before. We …rst solve for the seller’soptimal price posting problem, taking z^ as given. Similar to the case with a uniform distribution, z is de…ned as 1 z = u(q1) u(qx)dx Ze0 where q1 remains the same as before and (e0; qx)x [e0;1] solve 2

e0f(e0) + F (e0) = 1; 1 F (e) e u0(qx) = c0(qx): f(e)

We characterize the solution in Lemma 2.

Lemma 2 For any given z^ z, the optimal solution (qe; ze; ve) for all e [0; 1] to the seller’s price 2 posting problem is unique and is characterized by

(i) For e [0; e0], qe = ze = ve = 0; 2 (ii) For e [e0; e^], 2

1 F (e) e^ qe : e u0(qe) = c0(qe); (20) f(e) [1 F (ê)]2 (ê) = ; e^ ef^ (ê) + 1 F (ê) ze = eu(qe) ve; e ve = u(qe)dF (e); Z0 9 In Nosal (forthcoming), the terms of trade in the DM is determined by buyers making take-it-or-leave-it o¤ers.

17 (iii) For e [^e; 1], 2

1 F (^e) e^ qe =q ^ : e^ u0(^q) = c0(^q); (21) f(^e) ze =z; ^

ve = eu(^q) z^;

(iv) e0 and e^ are given by

e^ eu^ (^q) u(qx)dx =z; ^ (22) Ze0 e0f(e0) + F (e0) 1 + e^ = 0: (23)

For z^ > z, (22) is replaced by e^ = 1.

Proof. See the Appendix. With a general continuous distribution, the pricing schedule posted by sellers is again to extract buyers’private information about their preference shocks. Low preference buyers optimally choose to hold on to their money balances and wait for future consumption opportunities. The endogenous extensive margin e¤ect still exists. Buyers with preferences higher than e0 but below e^ spend part of their money balances and consume more as e increases. For buyers with [^e; 1], they spend all their money. We also have the following comparative statics results.

Proposition 5 The seller’s optimal pricing schedule has the following properties: de=d^ z^ > 0, dq=d^ z^ > 0, de0=dz^ > 0 and dqe=dz^ < 0 for e [e0; e^]. 2

Proof. See the appendix.

Proposition 6 Quantity discounts: when c(q) = q, d(ze=qe)=dqe < 0 for e [e0; e^]. 2

Proof. See the appendix.

5.2 Equilibrium

After deriving the seller’s pricing schedule in the DM, we can solve the buyer’s demand for money in the CM from (18). The de…nition of a price posting equilibrium remains the same as De…nition 1.

Since S(^z) and S0(^z) do not directly depend on i, a unique monetary equilibrium exists for generic values of . Recall that ef(e) is increasing in e. From (23), one can show that 1 e^ < ef^ (^e) + F (^e) and hence e0 < e^. There is a continuum of prices observed in equilibrium, which con…rms that the

18 law of two prices in Curtis and Wright (2004) cannot be generalized to a divisible money framework with a general continuous distribution.

Proposition 7 The e¤ects of in‡ation on (^z; e0; e^): dz=d ^ < 0; de0=d < 0 and de=d ^ < 0.

Proof. See the appendix.

Proposition 8 The e¤ects of in‡ation on (qe; ze; ze=z^) for e [e0; e^]: dqe=d > 0, dze=d > 0, and 2 d(ze=z^)=d > 0 if u000 0 and c000 0 and f 0(e) 0:

Proof. See the appendix. In‡ation reduces buyers’choice of the real money balance. In‡ation also induces more buyers to spend their money. The model with a more general distribution can still capture the hot potato e¤ect along the extensive margin. To show the hot potato e¤ect along the intensive margin, we impose the assumption f 0(e) 0. To summarize, most of our results from a uniform distribution of preference shocks remain valid when the distribution of preference shocks is extended to a more general continuous distribution.

It shows that price posting with undirected search is an interesting pricing mechanism to examine because it captures several commonly observed pricing behaviors such as quantity discounts and the hot potato e¤ect of in‡ation. It also veri…es that the divisibility of money matters. The law of two prices found in the indivisible framework no longer holds when money becomes divisible.

6 Conclusion

This paper studies the monetary equilibrium when sellers post prices and search is undirected. It contributes to the literature by considering a continuous distribution of the match speci…c preference shocks in a divisible money framework. We show that price posting monetary equilibrium exists and it is unique. Unlike the predictions from the indivisible money framework or the divisible money framework with a two-point distribution of preference shocks, we …nd that there exists a continuum of price-quantity pairs and the law of two prices does not hold. The model can also be used to explain seller’spricing behavior and the buyer’sspending pattern. The equilibrium pricing schedule features quantity discounts, a common pricing practice. The model o¤ers a very natural explanation of the hot potato e¤ect –buyers spend money faster when facing higher in‡ation. As in‡ation increases, the value of retaining money for future spending decreases. So more buyers start spending money and in addition, buyers spend a larger fraction of their money.

19 We think that it is useful to study price posting because it is more realistic than bargaining or price taking in many situations. One may argue that it is hard to categorize whether search is directed or undirected in the real world. In the future, we plan to study price posting with a mixture of directed search (informed buyers) and undirected search (uninformed buyers). It will be interesting to investigate how informed buyers interact with uninformed buyers and how sellers set prices in this context.

20 A Appendix

A.1 Proof of Proposition 6: Seller’sOptimal Price Posting with a General Distribution

Proof. The seller’sproblem is

1 max [ c(qe) + eu(qe) ve]dF (e) qe;ve Z0

qe 0; (NN) 8 > > eu(qe) ve z^; (CC) > > s.t. > > @qe=@e 0; (IC1) > <> v e = u(qe); (IC2) > > > > v = 0. (PC) > 0 > > :> for all e [0; 1]. 2 To proceed, we …rst ignore the constraint @qe=@e 0. We will impose the constraint later to …nd the solution. The Hamiltonian of the optimal control problem is H = [ c(qe) + eu(qe) ve]f(e) + eu(qe) where e is the co-state variable. The Lagrangian is

= f(e)[eu(qe) c(qe) ve] + eu(qe) L

+ [^z eu(qe) + ve] e

+eqe;

where e and e are the Lagrangian multipliers associated with the cash constraint and the non- negativity constraint, respectively. The …rst-order conditions are

qe :[ef(e) + e e] u0(qe) = f(e)c0(qe) e; (24) e : ve : f(e) + = e; (25) e : e : u(qe) = ve; (26)

:z ^ eu(qe) + ve 0; if > 0, then = 0; (27) e e

e : qe 0, if > 0, then e = 0. (28)

21 The transversality condition is 1v1 = 0. For monetary equilibrium to exist, v1 > 0 and 1 = 0. Integrating (25) over the interval [e; 1], we have

1 : 1 xdx = F (1) F (e) dx x Ze Ze

1 and 1 e = 1 F (e) e;where e = dx. Since we focus on monetary equilibria, it follows e x from the transversality condition that R

e = F (e) 1 + e: (29)

Plugging (29) into (24), we get

[ef(e) + F (e) 1 + e e] u0(qe) = f(e)c0(qe) e: (30) e

Since @qe=@e 0 (and as a result @ze=@e 0), we can discuss the solution to the seller’sproblem by dividing e into three regions characterized by two threshold values of e, e0 and e^. We will consider three cases in the following.

Case (a). For e e0, the NN binds and qe = ze = ve = 0. 10 Case (b). For e [e0; e^], neither NN nor CC binds or = e = 0. (30) reduces to 2 e

[ef(e) + F (e) 1 + e^] u0(qe) = f(e)c0(qe) (31)

Case (c). For e [^e; 1], CC binds. In this case, ze =z ^, qe =q ^ with q^ solving 2

[êf(ê) + F (ê) 1 + e^] u0(^q) = f(ê)c0(^q) (32)

In the next step, we solve for e^ as a function of e^. To do this, note that for e (^e; 1), CC binds 2 and qe =q ^ solves

[ef(e) + F (e) 1 + e e] u0(^q) = f(e)c0(^q) (33) e

Combining (32) and (33), we reach

ef^ (ê)f(e) + F (ê)f(e) f(e) + f(e)e^ (34) : = ef(e)f(ê) + F (e)f(ê) f(ê) + f(ê)e + eef(ê). 10 Notice that e = e^ for e [0; e^]. 2

22 : where we use = e. Rewriting (34) as e

: 1 F (ê) 1 e^ f(e) F (e) 1 e + e = e^ + + f(e) + Q(e); e f(ê) f(ê) f(ê) e e e which is a …rst-order di¤erential equation of e. Solving the …rst-order di¤erential equation, we have

1 = eQ(e)de: e e Z and

F (ê) 1 e^ ee = e^ + + F (e) eF (e) + e + C (35) f(ê) f(ê) f(ê) where C is a constant determined by the boundary condition 1 = 0:

1 F (ê) C = e^ e^ : f(ê) f(ê) f(ê)

We can solve e^ as a function of e^ by using (35) for e =e ^:

[1 F (ê)]2 1 F (ê) e^(ê) = = < 1 F (ê): (36) ef^ (ê) + 1 F (ê) ef^ (ê) 1 F (ê) + 1

Since ef(e) increases in e, the numerator of e^ increases in e^. It follows that e^ decreases in e^.

Finally, we will prove that @qe=@e > 0 for e [e0; e^] to justify that the solution satis…es the 2 constraint @qe=@e 0. We rearrange the …rst-order condition for qe [e0;e^] as 2

u (q ) f(e) 1 0 e = = e : F (e) 1 c0(qe) ef(e) + F (e) 1 + e^ 1 + + e^ ef(e) ef(e)

1 F (e) 1 e^ As e is decreasing in e, we only need to show that (e) = ef(e) + ef(e) is increasing in e. Since

e^ < 1 F (^e) and f(e) + ef 0(e) > 0,

2 e[f(e)] + [1 F (e) e^][f(e) + ef 0(e)] 0(e) = > 0: [ef(e)]2

Using the results above, the derivation of Proposition 6 is a straightforward exercise.

23 A.2 Proof of Proposition 7: Properties of Seller’sOptimal Pricing Sched- ule

1 F (ê) e^ c0(^q) Proof. Let ' (ê) e^ , (32) can be rewritten '(ê) = . Di¤erentiate it with respect f(ê) u0(^q) to e^ and we have dq^ ' (ê)(u )2 = 0 0 de^ c u c u 00 0 0 00 Use (36) to rearrange '(ê) as e^ ' e : (^) = 1 F (ê) 1 + ef^ (ê)

It is easy to show that '0(^e) > 0. If then follows that dq=d^ e^ > 0: Di¤erentiate (23) with respect to e^ and we have

de de^ 0 = de^ > 0: de^ 2f(e0) + ef(e0)

Di¤erentiating (20) with respect to e^ gives

1 d dq e^ e = f(e) de^ < 0: c00u0 c0u00 de^ 2 u0

Di¤erentiate (22) with respect to z^,

e^ dq^ de^ de^ dqx eu^ 0(^q) u0(qx) dx = 1; de^ dz^ dz^ de^ Ze0 from which we derive de^ 1 = e^ dz^ dq^ dqx eu^ 0(^q) u0(qx) dx de^ e0 de^ R Since dq=d^ e^ > 0, dqe=de^ < 0 for e [e0; e^], we have de=d^ z^ > 0: It then follows that dq=d^ z^ > 0, 2 de0=dz^ > 0 and dqe=dz^ < 0 for e [e0; e^]. 2

24 A.3 Proof of Proposition 8: Quantity Discount with General Distribution

Proof. For e [e0; e^], qe is solved from (20). Integrating both sides from e0 to e, we have 2

e e 1 F (x) e^ x u0(qx)dx = c0(qx)dx f(x) Ze0 Ze0 e 2 1 F (e) e^ [f(x)] [1 F (x) e^]f 0(x) e u(qe) u(qx) 1 dx = c(qe) ! f(e) [f(x)]2 Ze0 e e 2 1 F (e) e^ [f(x)] + [1 F (x) e^]f 0(x) u(qx)dx = e u(qe) u(qx) dx c(qe): ! f(e) [f(x)]2 Ze0 Ze0

The associated money payment is

e ze = eu(qe) u(qx)dx Ze0

The real unit price is e z eu(qe) u(qx)dx e = e0 qe Rqe

Di¤erentiating ze=qe with respect to qe, we have

ze @e @e e @ u(qe) + eu0(qe) u(qe) qe eu(qe) u(qx)dx qe @qe @qe e0 = 2 @q e h qei h R i e eu0(qe)qe eu(qe) + u(qx)dx e0 = 2 qe R e eu0(qe)qe eu(qe) + u(qx)dx _ Ze0 1 F (e) e^ = eu0(qe)qe eu(qe) + e u(qe) f(e) e 2 [f(x)] + [1 F (x) e^]f 0(x) u(qx) dx c(qe) [f(x)]2 Ze0 1 F (e) e^ = eu0(qe)qe u(qe) f(e) e 2 [f(x)] + [1 F (x) e^]f 0(x) u(qx) dx c(qe) [f(x)]2 Ze0 1 F (e) e^ = [u0(qe)qe u(qe)] + c0(qe)qe f(e) e 2 [f(x)] + [1 F (x) e^]f 0(x) u(qx) dx c(qe) [f(x)]2 Ze0 where "_" represents "have the same sign as".

If c(qe) is linear, c0(qe)qe = c(qe). We have shown earlier that e^ < 1 F (e) or 1 F (e) e^ > 0. Since u00 < 0, we have u0(qe)qe u(qe) < 0: To prove the quantity discounts result, we only need to

25 2 show that [f(x)] + [1 F (x) e^]f 0(x) > 0.

2 [f(x)] + [1 F (x) e^]f 0(x) > 0 is equivalent to 2 [f(x)] [1 F (x) e^]f 0(x) x + x > 0 or f(x) f(x)

[1 F (x) e^]f 0(x) xf(x) + x > 0 f(x)

1 F (x) e^ Using x > f(x) > 0 for x > e0 (note the …rst-order condition with respect to qe), we show that

[1 F (x) e^]f 0(x) xf(x) + x f(x)

[1 F (x) e^]f 0(x) > 1 F (x) e^ + x f(x)

xf 0(x) = [1 F (x) e^] 1 + f(x) f(x) + xf 0(x) = [1 F (x) e^] f(x) > 0 if xf(x) increases in x:

A.4 Proof of Proposition 9: E¤ects of In‡ation

Proof. The proof is similar to the proof of Proposition 4. As S0(z) = 0, one can show that

S0(z) + 1 < 0 for > and from above. We again focus on z^ [0; z]. For a generically ! 2 dz^ 1 unique z^, it must be true that S00(^z) < 0. So = < 0. The rest of the results follow from d S00(^z) Proposition 7.

A.5 Proof of Proposition 10: Hot Potato E¤ect with General Distribution

Proof. As de=d ^ < 0, proving dqe=d > 0 and dze=d > 0 is equivalent to proving dqe=de^ < 0 and

dqe dze dze=de^ < 0. We do this by proving d de =de^ < 0 and d de =de^ < 0, or qe(e;e ^) and ze(e;e ^) are steeper for lower e^.

dqe We …rst show d =de^ < 0. Use equation (20) that characterizes qe for e [e0; e^] to calculate de 2 dqe=de as (to simplify notation, we omit the arguments of the functions u( ) and c( )):

2 2 dqe u0 f + (1 F e^]f 0) = 1 + : de c u c u f 2 00 0 0 00

26 2 u0 The …rst term c u c u > 0 decreases in e^ if c000 0 and u000 0: 00 0 0 00

2 f +[1 F e^]f 0 The second term 1 + 2 > 0 decreases in e^ if f 0 0: f

2 f +(1 F e^)f 0 d 1 + f 2 de^=de^ = f 0 h de^ i f 2

0 if f 0 0.

dqe Since both the …rst term and the second term of dqe=de (are positive and) decrease in e^, d de =de^ < 0.

dze Now we show d de =de^ < 0. Remember that ze = eu0(qe)dqe=de, so 3 2 dze eu0 f + (1 F e^f 0) = 1 + de c u c u f 2 00 0 0 00

3 eu0 The …rst term c u c u > 0 decreases in e^ if c000 0 and u000 0: 00 0 0 00

The second term is positive and decreases in e^ if f 0 0. Since both the …rst term and the second dze term of dze=de (are positive and) decrease in e^, d de =de^ < 0.

As dze=d > 0 and dz=d ^ < 0, d(ze=z^)=d > 0.

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